The Kim-Pillay theorem for abstract elementary categories

We introduce the framework of AECats (abstract elementary categories), generalizing both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory (cat, as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim-Pillay theorem for first-order logic characterizes simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for AECats with the amalgamation property, generalizing the first-order version and existing versions for positive logic.